${\mathrm{GL}}_2\times {\mathrm{GSp}}_2$ $L$-values and Hecke eigenvalue congruences

Abstract

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as ${\mathrm{GSp}}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{A})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" $L$-value associated to cuspidal automorphic representations of ${\mathrm{GL}}_2(\mathbb{A})$ and ${\mathrm{GSp}}_2(\mathbb{A})$. Using special techniques for evaluating $L$-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.

To appear in Journal de Théorie des Nombres de Bordeaux. 21 pages